Foundations — Circuits and superposition

This page builds every symbol the parent note uses, starting from "what is a number in a network" and ending at "what does $s\sqrt{mp}$ mean". Nothing below assumes you have seen linear algebra. Read top to bottom — each block earns the next.

Two shorthand symbols we will lean on repeatedly, defined before first use:

0. The picture we keep coming back to

Everything in this topic lives inside activation space: a set of numbers a layer produces, drawn as a point (or an arrow from the origin) in a grid.

Keep this drawing in your head. Every symbol below is a label on this picture.

1. Neuron and activation — the raw numbers

The subscript $j$ is just a name tag: $a_1$ is neuron 1, $a_2$ is neuron 2, and so on. When we write $a_j$ with a letter, we mean "any one of them".

Where do these numbers come from? A neuron sums up its inputs and then passes them through an activation function (often ReLU, which zeros out negatives). That non-linear squash is why a neuron can stay silent for most inputs — a fact that matters later for sparsity.

2. The vector — a bundle of dials at once

So a neuron is one axis; a vector is one arrow built from readings along every axis. The parent note writes feature vectors like ${\mathbf{v}}_2=\left[\begin{array}{c}0.5\\ 0.866\end{array}\right]$ — that is just "0.5 right, 0.866 up".

The tiny subscript on the whole bold letter (${\mathbf{v}}_2$) names which arrow. The subscript on a plain letter ($v_2$ or $v_{ij}$) names which component. Same style, different job — watch which one is bold.

3. Dimension $n$ — how many axes the room has

4. Feature and feature count $m$ — the ideas we want to store

Here is exactly that case: $m=4$ feature arrows squeezed into an $n=2$ room.

Notice in the figure that only two arrows could ever be at perfect right angles in 2D — yet four fit if we let them lean slightly toward each other. That lean is the whole subject; we quantify it with the symbol $s$ in section 7.

The parent writes ${\mathbf{v}}_i\in {\mathbb{R}}^n$. Read $\in {\mathbb{R}}^n$ as "is a list of $n$ real numbers" — i.e. "${\mathbf{v}}_i$ is an arrow living in the $n$-axis room". $\mathbb{R}$ just means "ordinary numbers" (positive, negative, fractions), and the superscript $n$ means "$n$ of them".

5. Length of a vector — the norm $\Vert \mathbf{v}\Vert$

Before two arrows can be compared for "which way they point", we need to measure how long one arrow is. That length is called the norm.

Check the example: ${\mathbf{v}}_2=\left[\begin{array}{c}0.5\\ 0.866\end{array}\right]$ has $\Vert {\mathbf{v}}_2\Vert =\sqrt{0.5^2+0.866^2}=\sqrt{0.25+0.75}=1$ — indeed a unit arrow.

The right panel shows what a whole set of unit arrows looks like: all tips sit on the unit circle (in 2D) or unit sphere (in higher $n$), so comparing them is purely about angle.

6. The dot product — measuring how much two arrows agree

Now we need a tool. The topic keeps asking "do two feature arrows point the same way?" We need one number that answers that. That number is the dot product.

The picture below turns that into a curve: as the angle $\theta$ grows from $0°$ to $180°$, the dot product slides from $+1$ (aligned) through $0$ (right angle) to $-1$ (opposite).

We picked the dot product and not, say, ordinary subtraction because subtraction gives a new arrow (still $n$ numbers), whereas we want one honest number saying "aligned or not". That's the dot product's whole job.

7. Orthogonal vs. almost-orthogonal — the heart of superposition

First, one more piece of notation, then the overlap symbol.

8. Sparsity $p$ — how often a feature is actually present

The pink bars in the figure are the active features; the faded ones are silent. Out of $m$ possible features, only a small fraction $p$ are lit — that is what "sparse" looks like.

9. Interference and SNR — why sparsity saves us

When several active features share overlapping directions, their small overlaps add up and pollute your reading of any one feature. That pollution is interference.

Rearranging that inequality gives the parent's capacity formula $m\lesssim \frac{{\text{SNR}}_{\min }^2}{p{s}^2}$ — sparser features (smaller $p$) let you store more. That connects directly to how sparse autoencoders try to un-mix superposed features by forcing sparse activations.

10. The exponential and its inverse — capacity and undoing

Now the two facts the parent asserts, each with an honest sketch.

So two random arrows overlap by only about $1/\sqrt{n}$, and the number you can pack before any pair breaks the threshold grows exponentially in $n$. These are the two statistical facts the parent leans on.

11. Circuit words — the computation side

These link out to attention head interpretability (which heads do the reading/writing) and feature visualization (seeing what a direction means), and feed forward into concept activation vectors (naming a direction as a concept).

Prerequisite map

Equipment checklist

Cover the right side and answer out loud before revealing.

Why does capacity grow like $\